Pólya urn models.

*(English)*Zbl 1149.60005
Text in Statistical Science. Boca Raton, FL: CRC Press (ISBN 978-1-4200-5983-0/hbk; 978-1-4200-5984-7/ebook). xi, 290 p. (2009).

This book deals with the subject of discrete probability via several modern developments in urn theory and its numerous applications.

One main point in the book is that much of the endeavor in the field of (discrete) probability has roots in urn models, or can be cast in the framework of urn models. Some classical problems, initially not posed as urn questions, turn out to have a very natural interpretation or representation in terms of urns. Some of these classical problems originated in gambling and such areas.

Chapter 1 is about basic discrete distributions and a couple of modern tools such as stochastic processes and exchangeability. To conform with the aim and scope of the book the author switch to the language of urns as quickly as possible. Realization of distributions is substantiated with urn arguments, and examples for all concepts are in terms of urns. Chapter 2 presents some classical probability problems. Some of these problems were not originally given as urn problems but most of them can be recast as such. Chapter 3 is about dichromatic Pólya urns as a basic discrete structure, growing in discrete time. Chapter 4 considers an equivalent view in continuous time obtained by embedding the discrete Pólya urn scheme in Poisson processes, an operation called poissonization, from which we obtain the Pólya process. Chapter 5 gives heuristical arguments to connect the Pólya process to the discrete urn scheme (depoissonization). Chapter 6 is concerned with several extensions and generalizations (multicolor schemes, random additions). Chapter 7 presents an analytic view, where functional equations for moment generating functions can be obtained and solved. Chapter 8 is about applications to random trees, the kinds that appear in computer science as data structures or models for analysis of algorithms. Chapter 9 is on applications in bioscience (evolution, phylogeny, competitive exclusion, contagion, and clinical trial) where other types of Pólya-like urns appear. Chapter 10 is for urns evolving by multiple ball drawing and applications. All the chapters have exercises at the end that range from easy to challenging along with solutions in the back of the book that often shed more light on the topics.

One main point in the book is that much of the endeavor in the field of (discrete) probability has roots in urn models, or can be cast in the framework of urn models. Some classical problems, initially not posed as urn questions, turn out to have a very natural interpretation or representation in terms of urns. Some of these classical problems originated in gambling and such areas.

Chapter 1 is about basic discrete distributions and a couple of modern tools such as stochastic processes and exchangeability. To conform with the aim and scope of the book the author switch to the language of urns as quickly as possible. Realization of distributions is substantiated with urn arguments, and examples for all concepts are in terms of urns. Chapter 2 presents some classical probability problems. Some of these problems were not originally given as urn problems but most of them can be recast as such. Chapter 3 is about dichromatic Pólya urns as a basic discrete structure, growing in discrete time. Chapter 4 considers an equivalent view in continuous time obtained by embedding the discrete Pólya urn scheme in Poisson processes, an operation called poissonization, from which we obtain the Pólya process. Chapter 5 gives heuristical arguments to connect the Pólya process to the discrete urn scheme (depoissonization). Chapter 6 is concerned with several extensions and generalizations (multicolor schemes, random additions). Chapter 7 presents an analytic view, where functional equations for moment generating functions can be obtained and solved. Chapter 8 is about applications to random trees, the kinds that appear in computer science as data structures or models for analysis of algorithms. Chapter 9 is on applications in bioscience (evolution, phylogeny, competitive exclusion, contagion, and clinical trial) where other types of Pólya-like urns appear. Chapter 10 is for urns evolving by multiple ball drawing and applications. All the chapters have exercises at the end that range from easy to challenging along with solutions in the back of the book that often shed more light on the topics.

Reviewer: Nicko G. Gamkrelidze (Moskva)